Three-dimensional coordinate transformation problem is the most frequent problem in photogrammetry, geodesy, mapping, geographical information science (GIS), and computer vision. To overcome the drawback that traditional solution of the problem based on rotation angles depends strongly on initial value of parameter, which makes the method ineffective in the case of super-large rotation angle, the paper adopts an unit quaternion to represent three-dimensional rotation matrix, then puts forward a quaternion-based iterative solution of the problem. The cases study shows that the quaternion-based solution has no dependence on the initial value of parameter and desirable result with fast speed. Thus it is valid for three-dimensional coordinate transformation of any rotation angle.

Quaternion-Based Iterative Solution of Three- Dimensional Coordinate Transformation Problem

Based on the Lagrangian extremum law with the constraint that rotation matrix is an orthonormal matrix, the paper presents a new analytical algorithm of weighted 3D datum transformation It is a stepwise algorithm Firstly, the rotation matrix is computed using eigenvalue-eigenvector decomposition Then, the scale parameter is computed with computed rotation matrix Lastly, the translation parameters are computed with computed rotation matrix and scale parameter The paper investigates the stability of the presented algorithm in the cases that the common points are distributed in 3D, 2D, and 1D spaces including the approximate 2D and 1D spaces, and gives the corresponding modified formula of rotation matrix The comparison of the presented algorithm and classic Procrustes algorithm is investigated, and an improved Procrustes algorithm is presented since that the classic Procrustes algorithm may yield a reflection rather than a rotation in the cases that the common points are distributed in 2D space A simulative numerical case and a practical case are illustrated

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Analytical algorithm of weighted 3D datum transformation using the constraint of orthonormal matrix

Rigid transformation including rotation and translation can be elegantly represented by a unit dual quaternion. Thus, a non-differential model of the Helmert transformation (3D seven-parameter similarity transformation) is established based on unit dual quaternion. This paper presents a rigid iterative algorithm of the Helmert transformation using dual quaternion. One small rotation angle Helmert transformation (actual case) and one big rotation angle Helmert transformation (simulative case) are studied. The investigation indicates the presented dual quaternion algorithm (QDA) has an excellent or fast convergence property. If an accurate initial value of scale is provided, e.g., by the solutions no. 2 and 3 of Zavoti and Kalmar (Acta Geod Geophys 51:245–256, 2016) in the case that the weights are identical, QDA needs one iteration to obtain the correct result of transformation parameters; in other words, it can be regarded as an analytical algorithm. For other situations, QDA requires two iterations to recover the transformation parameters no matter how big the rotation angles are and how biased the initial value of scale is. Additionally, QDA is capable to deal with point-wise weight transformation which is more rational than those algorithms which simply take identical weights into account or do not consider the weight difference among control points. From the perspective of transformation accuracy, QDA is comparable to the classic Procrustes algorithm (Grafarend and Awange in J Geod 77:66–76, 2003) and orthonormal matrix algorithm from Zeng (Earth Planets Space 67:105, 2015. https://doi.org/10.1186/s40623-015-0263-6
 ).

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A dual quaternion algorithm of the Helmert transformation problem

The rigid motion involving both rotation and translation in the 3D space can be simultaneously described by a unit dual quaternion. Considering this excellent property, the paper constructs the Helmert transformation (seven-parameter similarity transformation) model based on a unit dual quaternion and then presents a rigid iterative algorithm of Helmert transformation using a unit dual quaternion. Because of the singularity of the coefficient matrix of the normal equation, the nine parameter (including one scale factor and eight parameters of a dual quaternion) Helmert transformation model is reduced into five parameter (including one scale factor and four parameters of a unit quaternion which can represent the rotation matrix) Helmert transformation one. Besides, a good start estimate of parameter is required for the iterative algorithm, hence another algorithm employed to compute the initial value of parameter is put forward. The numerical experiments involving a case of small rotation angles i.e. geodetic coordinate transformation and a case of big rotation angles i.e. the registration of LIDAR points are studied. The results show the presented algorithms in this paper are correct and valid for the two cases, disregarding the rotation angles are big or small. And the accuracy of computed parameter is comparable to the classic Procrustes algorithm from Grafarend and Awange (J Geod 77:66–76, 2003), the orthonormal matrix algorithm from Zeng (Earth Planets Space 67:105, 2015), and the algorithm from Wang et al. (J Photogramm Remote Sens 94:63–69, 2014).

Iterative solution of Helmert transformation based on a unit dual quaternion

The analytical solution of 3D datum transformation with an isotropic weight has been elegantly presented based on Procrustes algorithm (singular value decomposition). But the existence of analytical solution of 3D datum transformation with a non-isotropic weight needs further investigation. Based on the Lagrangian extremum law, the paper derives the analytical formula for translation parameter and scale factor, but because the rotation matrix is unsolved, the analytical solution does not exist. For this reason, the paper presents two kinds of iterative approach of 3D datum transformation with a non-isotropic weight. One is the iterative approach dependent on the objective function value, which uses the Lagrangian minimum function in the variable of rotation matrix as the objective function, and the other is the iterative approach dependent on the derivative of function, which uses the 3D datum transformation model that eliminates the translation parameter. In order to improve the speed and reliability of iterative computation, the form of rotation matrix represented by Rodrigues matrix instead of rotation angles or unit quaternion is adopted for the two iterative approaches. A numerical experiment is demonstrated, and comparison analysis of the two iterative approaches is carried out. The result shows from the view of computing speed and reliability, the iterative approach based on derivatives is preferred.

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Iterative approach of 3D datum transformation with a non-isotropic weight

Nonlinear geodetic datum transformation is particular suitable for the case of large rotation angles, and has been an active area of research in recent years. In this paper, we propose a new analytical solution. The method expresses the scale parameter and translation parameters as the function of rotation matrix, through the derivation process. Then, according to the orthogonality of rotation matrix, the method adopts Rodrigues matrix to represent it. Using the relative formula of Rodrigues matrix with anti-symmetric matrix which is formed by a rotation vector, we derive the solution formula of the rotation vector. Finally, we obtain the analytical solution to the seven parameters. The case studies indicate that the proposed method is valid, regardless of whether the coordinates are polluted by noise or not, and whether rotation angles are big or small.

A new analytical solution of nonlinear geodetic datum transformation

Single-image space resection is a fundamental issue of photogrammetry. This paper designs an anti-symmetric matrix to constitute Rodrigues matrix, used to represent the rotation matrix, then establishes the mathematical model of space resection applied to pattern search method, further uses pattern search method to directly solve the three elements of the antisymmetric matrix, finally obtains the attitude elements and line elements. The algorithm needs only objective function value without linearization, and has weak demand of initial values of 3 elements of the anti-symmetric matrix; no matter the attitude angles are small or large. Illustrative examples are given.

Qinglin Yi

Papers

A new analytical solution of nonlinear geodetic datum transformation

A solution to space resection based on pattern search method